diff --git a/CHANGELOG.md b/CHANGELOG.md
index 83b3ab874f2a..15c5633c62b1 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -42,6 +42,8 @@ Remember to align the itemized text with the first line of an item within a list
 * Bug fixes
   * Fixed a bug that meant that negative static_argnums to a jit were mishandled
     by the jit dispatch fast path.
+  * Fixed a bug that meant triangular solves of batches of singular matrices
+    produce nonsensical finite values, instead of inf or nan (#3589, #15429).
 
 ## jax 0.4.30 (June 18, 2024)
 
diff --git a/tests/linalg_test.py b/tests/linalg_test.py
index 2a64b95b9452..dd0ae38d9aa4 100644
--- a/tests/linalg_test.py
+++ b/tests/linalg_test.py
@@ -16,6 +16,7 @@
 
 from functools import partial
 import itertools
+import unittest
 
 import numpy as np
 import scipy
@@ -33,6 +34,7 @@
 from jax._src.lax import linalg as lax_linalg
 from jax._src import test_util as jtu
 from jax._src import xla_bridge
+from jax._src.lib import xla_extension_version
 from jax._src.numpy.util import promote_dtypes_inexact
 
 config.parse_flags_with_absl()
@@ -1623,6 +1625,15 @@ def testTriangularSolveGradPrecision(self):
         (a, b),
         (a, b))
 
+  @unittest.skipIf(xla_extension_version < 277, "Requires jaxlib > 0.4.30")
+  def testTriangularSolveSingularBatched(self):
+    x = jnp.array([[1, 1], [0, 0]], dtype=np.float32)
+    y = jnp.array([[1], [1.]], dtype=np.float32)
+    out = jax.lax.linalg.triangular_solve(x[None], y[None], left_side=True)
+    # x is singular. The triangular solve may contain either nans or infs, but
+    # it should not consist of only finite values.
+    self.assertFalse(np.all(np.isfinite(out)))
+
   @jtu.sample_product(
     n=[1, 4, 5, 20, 50, 100],
     batch_size=[(), (2,), (3, 4)] if scipy_version >= (1, 9, 0) else [()],